- #1
neelakash
- 511
- 1
Homework Statement
I found the following problem in two places.But I doubt the first one is wrong.
Let [tex]\ u_1(\ x )[/tex] and [tex]\ u_2(\ x )[/tex] are two degenerate eigenfunctions of the hamiltonian [tex]\ H =\frac{\ p^2 }{2\ m }\ + \ V (\ x )[/tex]
Then prove that
[tex]
\int u_1(x)\left(x\frac{\hbar}{i}\frac{\partial}{\partial x}-\frac{\hbar}{i}\frac{\partial}{\partial x}x\right)u_2(x)dx=0.
[/tex]
and [tex]\ u_1 (\ x )[/tex] and [tex]\ u_2 (\ x )[/tex] are orthogonal to each other.
OR prove that
[tex]
\int u_1*(x)\left(x\frac{\hbar}{i}\frac{\partial}{\parti al x}+\frac{\hbar}{i}\frac{\partial}{\partial x}x\right)u_2(x)dx=0.
[/tex]
Homework Equations
The Attempt at a Solution
Now I do not know what is correct expression.I found them in different places.May be both are correct.
But in the first one the sandwitched operator (xp-px) is skew hermitian.Hence, its eigenvalues are imaginary!So I doubt about its validity.
Now how to prove?As xp+px is hermitian, it is tempting to think that if [tex]\ u_1 (\ x )[/tex] and [tex]\ u_2 (\ x )[/tex]
are non-degenerate eigenfunctions of this operator,then it is done.
But I wonder, how to show it?